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# Open Physics

### formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina

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Volume 15, Issue 1

# Study on maintaining formations during satellite formation flying based on SDRE and LQR

Zhang Ke
• School of Astronautics, Northwestern Polytechnical University, Xi’an, P.R. China
• National Key Laboratory of Aerospace Flight Dynamics, Xi’an, P.R. China
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/ He Zhenqi
• Corresponding author
• School of Astronautics, Northwestern Polytechnical University, Xi’an, P.R. China
• National Key Laboratory of Aerospace Flight Dynamics, Xi’an, P.R. China
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• Other articles by this author:
/ Lv Meibo
• School of Astronautics, Northwestern Polytechnical University, Xi’an, P.R. China
• National Key Laboratory of Aerospace Flight Dynamics, Xi’an, P.R. China
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• Other articles by this author:
Published Online: 2017-06-14 | DOI: https://doi.org/10.1515/phys-2017-0043

## Abstract

Due to the influence of various perturbations of space, satellites flying in formation cannot maintain specific configurations for long durations [1, 2]. In order to ensure that formation configurations are able to meet the requirements of space missions, it is important to maintain control of formation configurations. This is an urgent problem to be solved. The traditional control method for controlling formations is based on the average orbit element, and uses the assumption that the average orbit element deviation and the instantaneous orbit element deviation are approximately equal. However, the continuous control system is more difficult to achieve in engineering practice. Using a LQR (linear quadratic regulator) optimal control algorithm and SDRE (state-dependent Riccati equation) optimal control algorithm to maintain the formation flying [3, 4]. Through simulation, it was found that when using the SDRE controller in the system transition process time is shorter than when the LQR controller is used, and fuel consumption is less for the SDRE controller than for the LQR controller.

Keywords: Formation flying; Riccati; Matlab; LQR; SDRE

PACS: 95.40.+s

## 1 Introduction

With the development of small satellite technology, research on a number of small satellite co-operation to achieve a common function has become one of the hot issues in the field of space research. Compared with a traditional single large satellite [5], small satellites have the advantages of low quality, low cost and high reliability. When a satellite is broken, we can replace the bad satellite base on orbit reconstruction, thus extending the life of the entire system. As a result, small satellite networks have received wide attention. Since the beginning of the 1990s, the United States has launched a “ION-F”, “TechSat-21” and other research programs [5].

A small satellite is smooth between collaborative working, because between their formation keep close, so its development promotes the research of the relative motion theory of the spacecraft [6]. In formation flying, the traditional control method is based on the average orbit element, and uses the assumption that the average orbit element deviation and the instantaneous orbit element deviation are approximately equal.However, taking into account the actual project in the continuous control is more difficult.Based on the dynamics of formation flying, two controllers are designed: an LQR controller based on a linear model, and an SDRE controller based on a nonlinear model.

## 2 Formation flying dynamics

The dynamic model of a satellite in orbit can be obtainedaccording to Kepler’s equation: $r→¨+μr3r→=0$(1)

$\stackrel{\to }{r}$ represents a radial vector from the Earth’s center of mass to the satellite, μ is the gravitational constant, μ = GM, with M as the Earth’s mass.

From Figure 1, the equations of motion of the lead satellite can be obtained as [7]: $r→¨c+μrc3r→c=0$(2)

Figure 1

Schematic diagram of the relationship between the leader-satellite and the following-satellite

The equations of motion of the following satellite are, then: $r→¨d+μrd3r→d=f$(3)

In equation(3), f is the sum of perturbation and control forces exerted on the satellite [8], Hypothesis n represents the leader satellite’s orbital angular velocity, $\stackrel{\to }{\rho }$ as the relative distance between the center of the leader satellite and the center of the following satellite. $\stackrel{\to }{\rho }=x\stackrel{\to }{i}+y\stackrel{\to }{j}+z\stackrel{\to }{k}.$ Then: $ρ→¨=rd→¨−rc→¨=μrc3rc→−μrd3rd→+f=(x¨−n2x+2ny˙)i→+(y¨−n2y−2nx˙−n˙x)j→+z¨k→$(4) $rd→¨−rc→¨=μrc−3[rc→−(1+2yrc+ρ2rc2)−32(ρ→+rc→)]+f$(5)

And from (4) and (5) we can obtain: $x¨=n2x−2ny˙−n˙y−μrc−3(1+2yrc+ρ2rc2)−32x+fxy¨=n2y−2nx˙−n˙x−μrc−3(1+2yrc+ρ2rc2)−32y+μrc−3[rc−(1+2yrc+ρ2rc2)−32rc]+fyz¨=−μrc−3[rc→−(1+2yrc+ρ2rc2)−32z+fz$(6)

When the distance between the leader satellite and the following satellite is far less than the distance between the center of the Earth and the center of the leader satellite, the nonlinear term in the equation can be simplified by $\left(1+\frac{2y}{{r}_{c}}+\frac{{\rho }^{2}}{{r}_{c}^{2}}{\right)}^{\frac{-3}{2}}.$ $(1+2yrc+ρ2rc2)−32=(1+2yrc+x2+y2+z2rc2)−32≈1−3yrc$(7)

Making a series of linearization for nonlinear equation(6), then: $x¨=−2ny˙−n˙y+fxy¨=3n2y+2nx˙+n˙x+fyz¨=−n2z+fz$(8)

In equation (8), $\stackrel{˙}{n}=-\frac{2\mu e\mathrm{sin}\theta }{{r}_{1}^{3}},\phantom{\rule{thinmathspace}{0ex}}n=\stackrel{˙}{\theta }=\frac{\sqrt{P\mu }}{{r}_{1}^{2}},\phantom{\rule{thinmathspace}{0ex}}{r}_{1}=\frac{P}{1+e\mathrm{cos}\theta },\phantom{\rule{thinmathspace}{0ex}}P=\alpha \left(1-{e}^{2}\right).$ α is the length of the semi major axis, θ represents the true anomaly and e indicates the eccentricity.

When the leader satellite runs in a circular orbit, then equation(8) can be simplified as follows: $x¨=−2ny˙+fxy¨=3n2y+2nx˙+fyz¨=−n2z+fz$(9)

## 3 Design of LQR controller design

LQR (linear quadratic regulator) is one of the earliest and most mature state space design methods in modern control theory [9]. The optimal control law of state linear feedback is obtained by LQR, and it is easy to achieve closed-loop optimal control [10].

The linear dynamic model of satellite formation flying is rewritten as a state-space expression. $x˙(x)=A(x)x+B(x)u$(10)

In equation(10), u represents the system control variables: $u={\left[\begin{array}{c}{f}_{x}\phantom{\rule{1em}{0ex}}{f}_{y}\phantom{\rule{1em}{0ex}}{f}_{z}\end{array}\right]}^{\mathrm{T}}={\left[\begin{array}{c}{u}_{x}\phantom{\rule{1em}{0ex}}{u}_{y}\phantom{\rule{1em}{0ex}}{u}_{z}\end{array}\right]}^{\mathrm{T}};$

x indicates the system state variable: $x=xyzVxVyVzT=xyzx˙y˙z˙T;$

A(x) is the state matrix of the system: $A(x)=03×3A1A2A3;$

In the matrix, $\begin{array}{}{A}_{1}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\end{array}$ $A2=00003n2000−n2,A3=0−2n02n00000.$

B(x) is a control matrix: $B(x)=03×3B1;$

In the matrix, $\begin{array}{}{B}_{1}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right].\end{array}$

Assume the altitude of the leader-satellite is 800 km, the orbital radius of the leader-satellite is rc = 7.2 × 105 m, the orbital angular velocity is n = 0.045 rad/sec = 2.5 × 10−4 rad/sec. At this time the state space expression is: $x˙y˙z˙x¨y¨z¨=Dxyzx˙y˙z˙+000000000100010001uxuyuz$(11)

In equation(11), $D=0001000000100000010000−5×10−4001.875×10−705×10−40000−6.25×10−8000rank(Wc)=BABA2BA3BA4BA5B=6$(12)

This indicates that the system can be controlled.

The objective function of LQR theory is: $Jc=12∫t0tf[(x−xd)TQ(x−xd)+uTRu]dt$(13)

In equation(13), xd is an ideal state, Q is used as the weight matrix of the error in the optimization process, and is a positive definite constant matrix of 6 × 6; R is the weight matrix of the control variables in the optimization process and is a positive semi-definite constant matrix of 3 × 3.

Solving Riccati equation: $−P˙=P(x)A(x)+AT(x)P(x)−P(x)B(x)R−1(x)P(x)+Q(x)$(14)

P(x) can be solved.

Feedback matrix: $K(x)=−R−1(x)BT(x)P(x)$(15)

The control law of the linear system in the performance index is: $u(x)=−K(x)(x−xd)$(16)

## 4 Design of SDRE controller design

The SDRE method is a kind of nonlinear control method. When applying this method, the nonlinear system dynamic equations must first be converted into SDC (State-dependent coefficients) forms [11].

For the nonlinear affine system: $x˙=f(x)+g(x)u$(17)

SDC forms can be obtained by pseudo linearization of nonlinear affine systems. $x˙=A(x)x+B(x)u$(18)

In equation(18): $x=x1x2x3x4x5x6T=xyzx˙y˙z˙T=uxuyuzT$

and $\begin{array}{}u={\left[\begin{array}{ccc}{u}_{x}& {u}_{y}& {u}_{z}\end{array}\right]}^{\text{\hspace{0.17em}T\hspace{0.17em}}}.\end{array}$

If the leader-satellite is running on a circular orbit, then $\stackrel{˙}{n}$ = 0.

At this time the state space expression is: $x˙1x˙2x˙3x˙4x˙5x˙6=Ax1x2x3x4x5x6+000000000100010001u1u2u3$(19)

In equation(19), $E=000100000010000001n2−μγ−n˙00−2n0n2+32μψrc4xn2−μγ+32μψrc3(2+yrc)32μψrc4z2n0000−μγ000$

rc is the orbital radius of the leader-satellite, rc = 7.2 × 105 m; the orbital angular velocity is n = 0.045 deg/sec = 2.5 × 104 rad/sec; μ is the geocentric gravitational constant, μ = 3.986 × 1014 m3/s2. $γ=((rc+x)2+y+z)32$(20) $ψ=1+ψ1+ψ2+⋅⋅⋅+ψn+⋅⋅⋅≈1+54ξ+3524ξ2+10564ξ3$(21)

In equation(20), $\xi =-\frac{2x}{{r}_{c}}-\frac{{x}^{2}+{y}^{2}+{z}^{2}}{{r}_{c}^{2}}.$

System controllability matrix: $Wc=A(x)A(x)B(x)A2(x)B(x)A3(x)B(x)A4(x)B(x)A5(x)B(x)$(22)

This calculation shows that the system is controlled point by point.

The objective function of SDRE theory is: $Jc=12∫t0∞[(x−xd)TQ(x)(x−xd)+uTR(x)u]dt$(23)

In equation(23), xd is an ideal state, Q(x) is used as the weight matrix of the error in the optimization process, and R(x) is used as the weight matrix of the control variables in the optimization process. Q(x) and R(x) are functions of the system state x.

Solving Riccati equation: $P(x)A(x)+AT(x)P(x)−P(x)B(x)R−1(x)P(x)+Q(x)$(24)

P(x) > 0 can be solved.

The control law for the nonlinear system in the performance index is: $u=−R−1(x)B(x)P(x)x$(25)

## 5 Numerical example and simulation results

Under ideal conditions, the initial relative position and velocity of the leader-satellite and the following-satellite formations are: $xd=0m,yd=170m,zd=0m,x˙d=0m/s,y˙d=−0.1m/s,z˙d=0m/s$

During a disturbance, the initial relative position and velocity error become: $x=20m,y=200m,z=−10m,x˙=0.3m/s,y˙=−0.3m/s,z˙=−0.1m/s$

So the initial relative position error and velocity error between the leader-satellite and the following-satellite are: xxd = [ 20 30 −10 0.3 0.2 −0.1]

In order to facilitate the comparison of the LQR and SDRE methods, we do not consider the influencing factors of the percussion force. the weight matrix Q and R are respectively: $Q=diag([111111]);R=diag([300300300])$

In the SDRE controller, the weight matrices Q and R are respectively: $Q(x)=diag(1+|x1|+|x2|1+|x2|1+|x3|1+|x4|1+|x5|1+|x6|);R(x)=diag(300+x13300+x23300+x34)$

Using MATLAB to get a variety of state simulations results are as follows:

## 6 Conclusions

Figures 2 to 4 compare the LQR controller algorithm and SDRE controller algorithm of the position error simulation. In 0-20 seconds, for the X-axis position error, the SDRE method tends to zero faster than the LQR. The curves for the Y- and Z-axes take at least 35 seconds to reach zero. Figures 5 to 7 compare the LQR and SDRE controller algorithms for the velocity error simulations. It can be seen that the SDRE method has less overshoot than the LQR method. From 0-7 seconds, for the X- and Y-axes, the SDRE method is faster than the LQR Method at tending to zero. From 7-35 seconds, the LQR method is faster than the SDRE method at tending to zero. But the Z-axis is the opposite. Figures 8 to 10 compare the LQR and SDRE controller algorithms of the control variables simulation . It can be seen that the range of control variables for the SDRE method is less than for the LQR method. At the same time, to reach steady state, the fuel consumed by the SDRE method is less than that of the LQR method.

Figure 2

X-axis position error change curve

Figure 3

Y-axis position error change curve

Figure 4

Z-axis position error change curve

Figure 5

X-axis velocity error change curve

Figure 6

Y-axis velocity error change curve

Figure 7

Z-axis velocity error change curve

Figure 8

X-axis control variables change curve

Figure 9

Y-axis control variables change curve

Figure 10

Z-axis control variables change curve

In summary, the simulation results show that, in the system transition, both processing time and fuel consumption are lower for the SDRE controller than for the LQR controller.

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Accepted: 2016-07-29

Published Online: 2017-06-14

Citation Information: Open Physics, Volume 15, Issue 1, Pages 394–399, ISSN (Online) 2391-5471,

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